The invention is based on an air flow rate measuring device as generally defined hereinafter. An air flow rate measuring device is already known (German Offenlegungsschrift 27 50 050 and corresponding U.S. Pat. No. 4,196,622), in which deposits occur on the surface of the temperature-dependent resistor embodied as a hot wire when the resistor is used for measuring the mass of air aspirated by an internal combustion engine; these deposits disadvantageously affect the precision of measurement and the speed of measurement. In this known air flow rate measuring device, a burnoff procedure is provided at specific times, for example, immediately following the shutoff of the ignition, during which process the deposits are burned off from the temperature-dependent resistor by means of an elevation in temperature of the resistor caused by a higher electrical current supplied to the resistor. However, it is unavoidable that a rich fuel-air mixture in the intake manifold will at some time be located in the vicinity of the temperature-dependent resistor and will ignite during the burnoff process at the red-hot temperature-dependent resistor. This unavoidable condition causes an undesirable further increase in temperature at the temperature-dependent resistor, which may cause the destruction of the resistor itself, particularly under the catalytic influence of a temperature-dependent resistor fabricated of platinum wire. The danger also exists that the combustion processes at the temperature-dependent resistor may last after the burnoff current through the resistor has been shut off, and a fresh fuel-air mixture may then be drawn into this combustion area surrounding the temperature-dependent resistor by a flue effect. Not only is there a danger that the temperature-dependent resistor may be destroyed as a consequence of such uncontrollable combustion, but there is also the danger that other parts of the engine will catch fire.